Interacting gauge fields and the zero-energy eigenstates in two dimensions

Gauge fields are formulated in terms of the zero-energy eigenstates of 2-dimensional Schrödinger equations with central potentials Va(ρ) = −agaρ (a 6= 0, ga > 0 and ρ = √ x2 + y2). It is shown that the zero-energy states can naturally be interpreted as a kind of interacting gauge fields of which effects are solved as the factors eigcχA , where χA are complex gauge functions written by the zero-energy eigenfunctions. We see that the gauge fields for a = 1 are nothing but tachyons that have negative squared-mass m2 = −g1. We also find out U(1)-type gauge fields for a = 1/2 and SU(3)-type gauge fields for a = 3/2. Massive particles with internal structures described by the zero-energy states are also studied. ∗E-mail: [email protected] It has been shown that 2-dimensional Schrödinger equations with the central potentials Va(ρ) = −agaρ (a 6= 0 and ρ = √ x2 + y2, i.e., x = ρ cosφ and y = ρ sinφ.) have zero-energy eigenstates which are infinitely degenerate [1, 2, 3, 4]. Let us briefly repeat the argument for driving the zero-energy states. The Schrödinger equations for the energy eigenvalue E , which are written as [− ~ 2 2m △ (x, y) + Va(ρ)] ψ(x, y) = E ψ(x, y) (1) where △ (x, y) = ∂/∂x + ∂/∂y, have zero-energy (E = 0) eigenstates. Note here that in this equation the mass m and the coordinate vector (x, y) can represent not only those of the single particle but those of the center of mass for a many particle system as well [4]. It has also been shown that, as far as the zero-energy eigenstates (ψ0) are concerned, the Schrödinger equations for all a can be reduced to the following equation in terms of the conformal transformations ζa = z a with z = x+ iy [2, 3]; [− ~ 2 2m △a − ga] ψ0(ua, va) = 0, (2) where △a = ∂ /∂ua + ∂ /∂v a, using the variables defined by the relation ζa = ua + iva, where ua = ρ a cos aφ and va = ρ a sin aφ. That is to say, the zero-energy eigenstates for all the different numbers of a are described by the same plane-wave solutions in the ζa plane. Furthermore it is easily shown that the zero-energy states degenerate infinitely. Let us consider the case for a > 0 and ga > 0. Putting the function f ± n (ua; va)e ±ikaua with ka = √ 2mga/~ into Eq. (2), where f ± n (ua; va) are polynomials of degree n (n = 0, 1, 2, · · · ), we obtain the equations for the polynomials [△a ± 2ika ∂ ∂ua ]f n (ua; va) = 0. (3) Note that from the above equations we can easily see the relation (f n (ua; va)) ∗ = f n (ua; va) for all a and n. General forms of the polynomials have been obtained by using the solutions in the a = 2 case (2-dimensional parabolic potential barrier) [2, 3]. Since all the solutions have the factors eaa or eaa , we see that the zero-energy states describe stationary flows [1, 2, 3]. Taking account of the direction of incoming flows in the ζa plane that is expressed by the angle α to the ua axis, the general eigenfunctions with zero-energy are written as arbitrary linear combinations of eigenfunctions included in two infinite series of { ψ ±(u) 0n (ua(α); va(α)) } for n = 0, 1, 2, · · · , where ψ ±(u) 0n (ua(α); va(α)) = f ± n (ua(α); va(α))e aa, (4) ua(α) = uacosα + vasinα, and 0 ≤ α < 2π. (For the details, see the sections II and III of Ref. [3].) It has been also pointed out that the motions of the z direction perpendicular to the xy plane can be introduced as free motions represented by ez. In this case the